Terminology for the use of datasets as data points
I don't think that this is anything new. Let's use your example of classifying an entire time series, say predicting word 1 vs word 2 for speech recognition. We can write out the data as a data frame like we would do with any other multivariate data: observations at time 1, time 2, etc as the predictors and the classification label as the response variable.

Each observation is a vector of the values at particular times for your subject, plus the label--no different than any other multivariate data. Sure, there might be special dependence structure because of the time series nature of your data, but you can still write it as a multivariate problem.

Okay, let's say that you hit the speech signal with a wavelet transform, resulting in an image-looking spectrogram of 2D data. Then just consider each "pixel" (time-frequency pair) to be a variable in your multivariate problem, along with the classification label. This is some kind of bijection between an $m\times n$ matrix and $\mathbb{R}^{n\times m}$.

You can extend this idea to 3D or 4D data (or higher), too. Just unwrap the high-dimension tensor in some kind of map $T^{m\times n \times \dots} \rightarrow \mathbb{R}^{m\times n \times \dots}$.